Anderson D.R., Sweeney D.J., Williams T.A. Essentials of Statistics for Business and Economics

1.2 Data

Data are the facts and figures collected, analyzed, and summarized for presentation and in-

terpretation. All the data collected in a particular study are referred to as the data set for the

study. Table 1.1 shows a data set containing information for 25 mutual funds that are part

of the Morningstar Funds 500 for 2008. Morningstar is a company that tracks over 7000

mutual funds and prepares in-depth analyses of 2000 of these. Its recommendations are fol-

lowed closely by financial analysts and individual investors.

Elements, Variables, and Observations

Elements are the entities on which data are collected. For the data set in Table 1.1, each in-

dividual mutual fund is an element: The element names appear in the first column. With 25

mutual funds, the data set contains 25 elements.

A variable is a characteristic of interest for the elements. The data set in Table 1.1 in-

cludes the following five variables:

• Fund Type: The type of mutual fund, labeled DE (Domestic Equity), IE (Interna-

tional Equity), and FI (Fixed Income)

• Net Asset Value ($): The closing price per share on December 31, 2007

1.2 Data 5

5-Year Expense

Fund Net Asset Average Ratio Morningstar

Fund Name Type Value ($) Return (%) (%) Rank

American Century Intl. Disc IE 14.37 30.53 1.41 3-Star

American Century Tax-Free Bond FI 10.73 3.34 0.49 4-Star

American Century Ultra DE 24.94 10.88 0.99 3-Star

Artisan Small Cap DE 16.92 15.67 1.18 3-Star

Brown Cap Small DE 35.73 15.85 1.20 4-Star

DFA U.S. Micro Cap DE 13.47 17.23 0.53 3-Star

Fidelity Contrafund DE 73.11 17.99 0.89 5-Star

Fidelity Overseas IE 48.39 23.46 0.90 4-Star

Fidelity Sel Electronics DE 45.60 13.50 0.89 3-Star

Fidelity Sh-Term Bond FI 8.60 2.76 0.45 3-Star

Gabelli Asset AAA DE 49.81 16.70 1.36 4-Star

Kalmar Gr Val Sm Cp DE 15.30 15.31 1.32 3-Star

Marsico 21st Century DE 17.44 15.16 1.31 5-Star

Mathews Pacific Tiger IE 27.86 32.70 1.16 3-Star

Oakmark I DE 40.37 9.51 1.05 2-Star

PIMCO Emerg Mkts Bd D FI 10.68 13.57 1.25 3-Star

RS Value A DE 26.27 23.68 1.36 4-Star

T. Rowe Price Latin Am. IE 53.89 51.10 1.24 4-Star

T. Rowe Price Mid Val DE 22.46 16.91 0.80 4-Star

Thornburg Value A DE 37.53 15.46 1.27 4-Star

USAA Income FI 12.10 4.31 0.62 3-Star

Vanguard Equity-Inc DE 24.42 13.41 0.29 4-Star

Vanguard Sht-Tm TE FI 15.68 2.37 0.16 3-Star

Vanguard Sm Cp Idx DE 32.58 17.01 0.23 3-Star

Wasatch Sm Cp Growth DE 35.41 13.98 1.19 4-Star

Source: Morningstar Funds 500 (2008).

TABLE 1.1

DATA SET FOR 25 MUTUAL FUNDS

file

W

EB

Morningstar

Data sets such as

Morningstar are

available on the

website for this text.

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• 5-Year Average Return (%): The average annual return for the fund over the past

5 years

• Expense Ratio: The percentage of assets deducted each fiscal year for fund expenses

• Morningstar Rank: The overall risk-adjusted star rating for each fund; Morningstar

ranks go from a low of 1-Star to a high of 5-Stars

Measurements collected on each variable for every element in a study provide the data. The

set of measurements obtained for a particular element is called an observation. Referring

to Table 1.1, we see that the set of measurements for the first observation (American Cen-

tury Intl. Disc) is IE, 14.37, 30.53, 1.41, and 3-Star. The set of measurements for the sec-

ond observation (American Century Tax-Free Bond) is FI, 10.73, 3.34, 0.49, and 4-Star, and

so on. A data set with 25 elements contains 25 observations.

Scales of Measurement

Data collection requires one of the following scales of measurement: nominal, ordinal,

interval, or ratio. The scale of measurement determines the amount of information con-

tained in the data and indicates the most appropriate data summarization and statistical

analyses.

When the data for a variable consist of labels or names used to identify an attribute of

the element, the scale of measurement is considered a nominal scale. For example, refer-

ring to the data in Table 1.1, we see that the scale of measurement for the Fund Type vari-

able is nominal because DE, IE, and FI are labels used to identify the category or type of

fund. In cases where the scale of measurement is nominal, a numerical code as well as non-

numerical labels may be used. For example, to facilitate data collection and to prepare the

data for entry into a computer database, we might use a numerical code by letting 1 denote

Domestic Equity, 2 denote International Equity, and 3 denote Fixed Income. In this case the

numerical values 1, 2, and 3 identify the category of fund. The scale of measurement is nomi-

nal even though the data appear as numerical values.

The scale of measurement for a variable is called an ordinal scale if the data ex-

hibit the properties of nominal data and the order or rank of the data is meaningful. For

example, Eastside Automotive sends customers a questionnaire designed to obtain data

on the quality of its automotive repair service. Each customer provides a repair service

rating of excellent, good, or poor. Because the data obtained are the labels—excellent,

good, or poor—the data have the properties of nominal data. In addition, the data can be

ranked, or ordered, with respect to the service quality. Data recorded as excellent indi-

cate the best service, followed by good and then poor. Thus, the scale of measurement

is ordinal. As another example, note that the Morningstar Rank for the data in Table 1.1 is

ordinal data. It provides a rank from 1 to 5-Stars based on Morningstar’s assessment of the

fund’s risk-adjusted return. Ordinal data can also be provided using a numerical code, for

example, your class rank in school.

The scale of measurement for a variable is an interval scale if the data have all the

properties of ordinal data and the interval between values is expressed in terms of a fixed

unit of measure. Interval data are always numerical. College admission SAT scores are an ex-

ample of interval-scaled data. For example, three students with SAT math scores of 620, 550,

and 470 can be ranked or ordered in terms of best performance to poorest performance in

math. In addition, the differences between the scores are meaningful. For instance, student 1

scored 620 ⫺ 550 ⫽ 70 points more than student 2, while student 2 scored 550 ⫺ 470 ⫽ 80

points more than student 3.

The scale of measurement for a variable is a ratio scale if the data have all the prop-

erties of interval data and the ratio of two values is meaningful. Variables such as dis-

tance, height, weight, and time use the ratio scale of measurement. This scale requires that

a zero value be included to indicate that nothing exists for the variable at the zero point.

6 Chapter 1 Data and Statistics

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For example, consider the cost of an automobile. A zero value for the cost would indicate

that the automobile has no cost and is free. In addition, if we compare the cost of $30,000

for one automobile to the cost of $15,000 for a second automobile, the ratio property

shows that the first automobile is $30,000/$15,000 ⫽ 2 times, or twice, the cost of the

second automobile.

Categorical and Quantitative Data

Data can be classified as either categorical or quantitative. Data that can be grouped by

specific categories are referred to as categorical data. Categorical data use either the nom-

inal or ordinal scale of measurement. Data that use numerical values to indicate how much

or how many are referred to as quantitative data. Quantitative data are obtained using

either the interval or ratio scale of measurement.

Acategorical variable is a variable with categorical data, and a quantitative variable

is a variable with quantitative data. The statistical analysis appropriate for a particular vari-

able depends upon whether the variable is categorical or quantitative. If the variable is

categorical, the statistical analysis is limited. We can summarize categorical data by count-

ing the number of observations in each category or by computing the proportion of the

observations in each category. However, even when the categorical data are identified by a

numerical code, arithmetic operations such as addition, subtraction, multiplication, and

division do not provide meaningful results. Section 2.1 discusses ways for summarizing

categorical data.

Arithmetic operations provide meaningful results for quantitative variables. For exam-

ple, quantitative data may be added and then divided by the number of observations to

compute the average value. This average is usually meaningful and easily interpreted. In

general, more alternatives for statistical analysis are possible when data are quantitative.

Section 2.2 and Chapter 3 provide ways of summarizing quantitative data.

Cross-Sectional and Time Series Data

For purposes of statistical analysis, distinguishing between cross-sectional data and time

series data is important. Cross-sectional data are data collected at the same or approxi-

mately the same point in time. The data in Table 1.1 are cross-sectional because they de-

scribe the five variables for the 25 mutual funds at the same point in time. Time series

data are data collected over several time periods. For example, the time series in

Figure 1.1 shows the U.S. average price per gallon of conventional regular gasoline be-

tween 2006 and 2009. Note that higher gasoline prices have tended to occur in the sum-

mer months, with the all-time-high average of $4.05 per gallon occurring in July 2008.

By January 2009, gasoline prices had taken a steep decline to a three-year low of $1.65

per gallon.

Graphs of time series data are frequently found in business and economic publications.

Such graphs help analysts understand what happened in the past, identify any trends over

time, and project future levels for the time series. The graphs of time series data can take

on a variety of forms, as shown in Figure 1.2. With a little study, these graphs are usually

easy to understand and interpret.

For example, Panel (A) in Figure 1.2 is a graph that shows the Dow Jones Industrial

Average Index from 1997 to 2010. In April 1997, the popular stock market index was near

7000. Over the next 10 years the index rose to over 14,000 in July 2007. However, notice

the sharp decline in the time series after the all-time high in 2007. By March 2009, poor

economic conditions had caused the Dow Jones Industrial Average Index to return to the

7000 level of 1997. This was a scary and discouraging period for investors. By January

2010, the index was showing a recovery by reaching 10,600.

1.2 Data 7

The statistical method

appropriate for

summarizing data depends

upon whether the data are

categorical or quantitative.

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The graph in Panel (B) shows the net income of McDonald’s Inc. from 2003 to 2009. The

declining economic conditions in 2008 and 2009 were actually beneficial to McDonald’s as

the company’s net income rose to an all-time high. The growth in McDonald’s net income

showed that the company was thriving during the economic downturn as people were cutting

back on the more expensive sit-down restaurants and seeking less-expensive alternatives

offered by McDonald’s.

Panel (C) shows the time series for the occupancy rate of hotels in South Florida over

a one-year period. The highest occupancy rates, 95% and 98%, occur during the months

of February and March when the climate of South Florida is attractive to tourists. In fact,

January to April of each year is typically the high-occupancy season for South Florida

hotels. On the other hand, note the low occupancy rates during the months of August to

October, with the lowest occupancy rate of 50% occurring in September. High tempera-

tures and the hurricane season are the primary reasons for the drop in hotel occupancy

during this period.

8 Chapter 1 Data and Statistics

Date

Mar 06 Oct 06 Apr 07 Nov 07 Jun 08 Dec 08 Jul 09

Average Price per Gallon

$0

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

$4.00

$4.50

FIGURE 1.1 U.S. AVERAGE PRICE PER GALLON FOR CONVENTIONAL

REGULAR GASOLINE

Source: Energy Information Administration, U.S. Department of Energy, July 2009.

NOTES AND COMMENTS

1. An observation is the set of measurements ob-

tained for each element in a data set. Hence, the

number of observations is always the same as the

number of elements. The number of measure-

ments obtained for each element equals the num-

ber of variables. Hence, the total number of data

items can be determined by multiplying the num-

ber of observations by the number of variables.

2. Quantitative data may be discrete or continuous.

Quantitative data that measure how many (e.g.,

number of telephone calls received in 15 min-

utes) are discrete. Quantitative data that measure

how much (e.g., weight or time) are continuous

because no separation occurs between the possi-

ble data values.

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FIGURE 1.2 AVARIETY OF GRAPHS OF TIME SERIES DATA

1.2 Data 9

Month

Percentage Occupied

20

40

60

80

100

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

(C) Occupancy Rate of South Florida Hotels

Year

1998 2000 2002 2004 2006 2008 2010

Dow Jones Industrial Average

5000

6000

7000

8000

9000

10000

11000

13000

12000

14000

(A) Dow Jones Industrial Avera

g

e

4

2

5

0

3

1

6

Net Income ($ billions)

Year

2003 2004 2005 2006 2007 2008 2009

(B) Net Income for McDonald’s Inc.

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1.3 Data Sources

Data can be obtained from existing sources or from surveys and experimental studies

designed to collect new data.

Existing Sources

In some cases, data needed for a particular application already exist. Companies maintain a

variety of databases about their employees, customers, and business operations. Data on

employee salaries, ages, and years of experience can usually be obtained from internal per-

sonnel records. Other internal records contain data on sales, advertising expenditures, distri-

bution costs, inventory levels, and production quantities. Most companies also maintain

detailed data about their customers. Table 1.2 shows some of the data commonly available

from internal company records.

Organizations that specialize in collecting and maintaining data make available sub-

stantial amounts of business and economic data. Companies access these external data

sources through leasing arrangements or by purchase. Dun & Bradstreet, Bloomberg, and

Dow Jones & Company are three firms that provide extensive business database services

to clients. ACNielsen and Information Resources, Inc., built successful businesses collecting

and processing data that it sells to advertisers and product manufacturers.

Data are also available from a variety of industry associations and special interest or-

ganizations. The Travel Industry Association of America maintains travel-related informa-

tion such as the number of tourists and travel expenditures by states. Such data would be of

interest to firms and individuals in the travel industry. The Graduate Management

Admission Council maintains data on test scores, student characteristics, and graduate man-

agement education programs. Most of the data from these types of sources are available to

qualified users at a modest cost.

The Internet continues to grow as an important source of data and statistical informa-

tion. Almost all companies maintain websites that provide general information about the

companyaswell asdataon sales, numberofemployees, number ofproducts,product prices,

and product specifications. In addition, a number of companies now specialize in making

information available over the Internet. As a result, one can obtain access to stock quotes,

meal prices at restaurants, salary data, and an almost infinite variety of information.

Government agencies are another important source of existing data. For instance, the

U.S. Department of Labor maintains considerable data on employment rates, wage rates,

sizeof thelabor force, andunion membership.Table1.3 listsselected governmentalagencies

10 Chapter 1 Data and Statistics

Source Some of the Data Typically Available

Employee records Name, address, social security number, salary, number of vacation days, num-

ber of sick days, and bonus

Production records Part or product number, quantity produced, direct labor cost, and materials cost

Inventory records Part or product number, number of units on hand, reorder level, economic

order quantity, and discount schedule

Sales records Product number, sales volume, sales volume by region, and sales volume by

customer type

Credit records Customer name, address, phone number, credit limit, and accounts receivable

balance

Customer profile Age, gender, income level, household size, address, and preferences

TABLE 1.2

EXAMPLES OF DATAAVAILABLE FROM INTERNAL COMPANY RECORDS

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and some of the data they provide. Most government agencies that collect and process data

also make the results available through a website. Figure 1.3 shows the homepage for the

U.S. Census Bureau website.

Statistical Studies

Sometimes the data needed for a particular application are not available through existing

sources. In such cases, the data can often be obtained by conducting a statistical study. Sta-

tistical studies can be classified as either experimental or observational.

In an experimental study, a variable of interest is first identified. Then one or more other

variables are identified and controlled so that data can be obtained about how they influence

the variable of interest. For example, a pharmaceutical firm might be interested in conducting

an experiment to learn about how a new drug affects blood pressure. Blood pressure is the

variable of interest in the study. The dosage level of the new drug is another variable that

is hoped to have a causal effect on blood pressure. To obtain data about the effect of the

1.3 Data Sources 11

Government Agency Some of the Data Available

Census Bureau Population data, number of households, and household

income

Federal Reserve Board Data on the money supply, installment credit, exchange

rates, and discount rates

Office of Management and Budget Data on revenue, expenditures, and debt of the federal

government

Department of Commerce Data on business activity, value of shipments by industry, level

of profits by industry, and growing and declining industries

Bureau of Labor Statistics Consumer spending, hourly earnings, unemployment

rate, safety records, and international statistics

TABLE 1.3

EXAMPLES OF DATA AVAILABLE FROM SELECTED GOVERNMENTAGENCIES

FIGURE 1.3 U.S. CENSUS BUREAU HOMEPAGE

The largest experimental

statistical study ever

conducted is believed to be

the 1954 Public Health

Service experiment for

the Salk polio vaccine.

Nearly 2 million children

in grades 1, 2, and 3 were

selected from throughout

the United States.

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new drug, researchers select a sample of individuals. The dosage level of the new drug is

controlled, as different groups of individuals are given different dosage levels. Before and

after data on blood pressure are collected for each group. Statistical analysis of the experi-

mental data can help determine how the new drug affects blood pressure.

Nonexperimental, or observational, statistical studies make no attempt to control the

variables of interest. Asurvey is perhaps the most common type of observational study. For

instance, in a personal interview survey, research questions are first identified. Then a ques-

tionnaire is designed and administered to a sample of individuals. Some restaurants use ob-

servational studies to obtain data about customer opinions on the quality of food, quality of

service, atmosphere, and so on. Acustomer opinion questionnaire used by Chops City Grill

in Naples, Florida, is shown in Figure 1.4. Note that the customers who fill out the ques-

tionnaire are asked to provide ratings for 12 variables, including overall experience, greet-

ing by hostess, manager (table visit), overall service, and so on. The response categories of

excellent, good, average, fair, and poor provide categorical data that enable Chops City Grill

management to maintain high standards for the restaurant’s food and service.

Anyone wanting to use data and statistical analysis as aids to decision making must be

aware of the time and cost required to obtain the data. The use of existing data sources is

desirable when data must be obtained in a relatively short period of time. If important data

are not readily available from an existing source, the additional time and cost involved in

obtaining the data must be taken into account. In all cases, the decision maker should

12 Chapter 1 Data and Statistics

Studies of smokers and

nonsmokers are

observational studies

because researchers do

not determine or control

who will smoke and who

will not smoke.

FIGURE 1.4 CUSTOMER OPINION QUESTIONNAIRE USED BY CHOPS CITY GRILL

RESTAURANT IN NAPLES, FLORIDA

Date: ____________ Server Name: ____________

Our customers are our top priority. Please take a moment to fill out our

survey card, so we can better serve your needs. You may return this card to the front

desk or return by mail. Thank you!

SERVICE SURVEY Excellent Good Average Fair Poor

Overall Experience ❑❑❑❑❑

Greeting by Hostess ❑❑❑❑❑

Manager (Table Visit) ❑❑❑❑❑

Overall Service ❑❑❑❑❑

Professionalism ❑❑❑❑❑

Menu Knowledge ❑❑❑❑❑

Friendliness ❑❑❑❑❑

Wine Selection ❑❑❑❑❑

Menu Selection ❑❑❑❑❑

Food Quality ❑❑❑❑❑

Food Presentation ❑❑❑❑❑

Value for $ Spent ❑❑❑❑❑

What comments could you give us to improve our restaurant?

Thank you, we appreciate your comments.—The staff of Chops City Grill.

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consider the contribution of the statistical analysis to the decision-making process. The cost

of data acquisition and the subsequent statistical analysis should not exceed the savings

generated by using the information to make a better decision.

Data Acquisition Errors

Managers should always be aware of the possibility of data errors in statistical studies. Using

erroneous data can be worse than not using any data at all. An error in data acquisition occurs

whenever the data value obtained is not equal to the true or actual value that would be ob-

tained with a correct procedure. Such errors can occur in a number of ways. For example, an

interviewer might make a recording error, such as a transposition in writing the age of a

24-year-old person as 42, or the person answering an interview question might misinterpret

the question and provide an incorrect response.

Experienced data analysts take great care in collecting and recording data to ensure that

errors are not made. Special procedures can be used to check for internal consistency of the

data. For instance, such procedures would indicate that the analyst should review the accu-

racy of data for a respondent shown to be 22 years of age but reporting 20 years of work

experience. Data analysts also review data with unusually large and small values, called

outliers, which are candidates for possible data errors. In Chapter 3 we present some of the

methods statisticians use to identify outliers.

Errors often occur during data acquisition. Blindly using any data that happen to be

available or using data that were acquired with little care can result in misleading informa-

tion and bad decisions. Thus, taking steps to acquire accurate data can help ensure reliable

and valuable decision-making information.

1.4 Descriptive Statistics

Most of the statistical information in newspapers, magazines, company reports, and other

publications consists of data that are summarized and presented in a form that is easy for

the reader to understand. Such summaries of data, which may be tabular, graphical, or

numerical, are referred to as descriptive statistics.

Refer again to the data set in Table 1.1 showing data on 25 mutual funds. Methods of

descriptive statistics can be used to provide summaries of the information in this data set.

For example, a tabular summary of the data for the categorical variable Fund Type is

shown in Table 1.4. A graphical summary of the same data, called a bar chart, is shown

in Figure 1.5. These types of tabular and graphical summaries generally make the data eas-

ier to interpret. Referring to Table 1.4 and Figure 1.5, we can see easily that the majority

of the mutual funds are of the Domestic Equity type. On a percentage basis, 64% are of

the Domestic Equity type, 16% are of the International Equity type, and 20% are of the

Fixed Income type.

1.4 Descriptive Statistics 13

Percent

Mutual Fund Type Frequency Frequency

Domestic Equity 16 64

International Equity 4 16

Fixed Income 5 20

Totals 25 100

TABLE 1.4

FREQUENCIES AND PERCENT FREQUENCIES FOR MUTUAL FUND TYPE

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A graphical summary of the data for the quantitative variable Net Asset Value, called

a histogram, is provided in Figure 1.6. The histogram makes it easy to see that the net as-

set values range from $0 to $75, with the highest concentration between $15 and $30. Only

one of the net asset values is greater than $60.

In addition to tabular and graphical displays, numerical descriptive statistics are used

to summarize data. The most common numerical descriptive statistic is the average, or

14 Chapter 1 Data and Statistics

0

Percent Frequency

Fund Type

Domestic Equity International Equity Fixed Income

10

20

30

40

50

60

70

FIGURE 1.5 BAR CHART FOR MUTUAL FUND TYPE

5

4

3

2

1

0

Frequency

Net Asset Value ($)

0

15 30 45 60 75

6

7

8

9

FIGURE 1.6 HISTOGRAM OF NETASSET VALUE FOR 25 MUTUAL FUNDS

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